Is there a section that covers finding the zeros of polynomials using p/q and synthetic division?

They now have added a section about synthetic division. Check it out here: https://www.khanacademy.org/math/algebra2/arithmetic-with-polynomials/synthetic-division-of-polynomials/v/synthetic-division I know my answer is a bit late but hopefully it may help others. Good luck!

Once you know whether the first interval is positive or negative, isn't it easy to tell what the rest of the intervals are based on the multiplicity of the zeros? The graph will always either touch or cross the x axis at every zero.

Correct. You could start with any interval and work away from it. If an interval includes zero, I find that easiest to check in factored form. Or if you also have the unfactored polynomial, the intervals on the ends might be obvious based on the leading term. But the methods in this article work even if you haven't learned about the behavior of multiple roots yet.

all of this is total shenanigans in the first skill for positive and negative intervals for polynomials it says that an interval can be both positive and negative when it is clearly only one. In the hints, it even looks like they changed the interval to fit their answer. Please explain I've already watched the video for this and the article spending many hours trying to figure this out.

*Positive interval*: The points for the function, or the graph sits above the x-axis *Negative interval*: The points for the function, or the graph sits below the x-axis If you have a graph, this is very easy - look at the graph and see if the line for the function sits above or below the x-axis. If you don't have a graph, then you need to test 1 or 2 values. -- If Y turns out to be positive, then the interval is positive because the point sits above the line. -- If Y turns out to be negative, then the interval is negative because the point sits below the line. For example, for the 1st problem to "Check Your Understanding", I used a value of x=-2 which is in the given interval. Plug in -2 for x in the function g(x)=(x+1)^2(x+6) g(-2) = (-2+1)^2(-2+6) = (-1)^2(4) = 1(4) = +4 The interval is positive because the point (-2, 4) is above the x-axis. Hope this helps.

i'm also confused by (no breaks in the graph)as the meaning of "continuous funtion" can anybody explaine to me more detail by "no breaks in the graph"?

It is exactly what it says. You will have a line/curve that extends from the left side to the right side of the coordinate plane, and there will be no gaps. Or, you can think of the equation / function as being defined for inputs values (x) = all real numbers. No values are excluded. An exclusion would leave a gap.

Could you explain how to solve this? The polynomial *p(x) = x^3 + 2x −11* has a real zero between which two consecutive integers? (A) 0 and 1 (B) 1 and 2 (C) 2 and 3 (D) 3 and 4 (E) 4 and 5

You could graph it and look at where the zeroes look like they are. visually you could test each of the options. the alternative is to factor it and find the zeroes in factored form. This one really doesn't look like it factors nicely, so I think it wants you to just graph it. If you don't have a graphing calculator you can find something online. Desmos is really good. After you graph it check between each interval. look on the graph and see if the line crosses between 0 and 1, then 1 and 2 and so on. Let me know if you still can't quite get it.

How do we figure out the maximum or minimum point the graph of polynomial will take?

To do this for an arbitrary polynomial, you need an understanding of calculus. I would suggest building up your trigonometry, geometry, and algebra background before diving into calculus!

The articles says in positive and negative intervals Why? _which means that the only way to change signs is to cross the x-axis. But if this happened, the given zeros would not be consecutive!_ I get that to change the sign the function needs to cross the x axis, but what does the latter part mean? Not consecutive anymore?

What the article is saying is this: A zero of a function is where the function touches the x axis. In order for the sign of a polynomial to change (positive to negative or negative to positive on the y axis), the polynomial needs to cross the x axis. Wherever the polynomial crosses the x axis, there will be a zero. Therefore, if two zeros are consecutive, the function cannot cross the x axis between them (because then there would be a zero in between those two zeros, which isn't possible because they are consecutive), and if the function cannot cross the x axis, the sign of the function cannot change.

When doing problems with positive and negative intervals of polynomials and it asks what the sign of f(x) is, given only the zeros (not the graph itself), how do we know if the graph starts from below the point or above the point? Because the graph could be drawn two ways right?

Read thru this article again and try the practice problems. Sal gives an example that has no graphs. You basically have to test values that fall between the x-intercepts. And, as Sal shows above, you don't even need to do the complete math. Once you pick the number you want to test, you can just see what sign that creates.

Main content

Course: Get ready for AP® Calculus > Unit 5

Lesson 7: Positive and negative intervals of polynomials

Positive & negative intervals of polynomials

Google Classroom

Learn about the relationship between the zeros of polynomials and the intervals over which they are positive or negative.

What you should be familiar with before taking this lesson

The zeros of a polynomial

f

correspond to the

x

-intercepts of the graph of

y = f (x)

For example, let's suppose

f (x) = (x + 3) (x - 1)^{2}

. Since the zeros of function

f

are

- 3

and

1

, the graph of

y = f (x)

will have

x

-intercepts at

(- 3, 0)

and

(1, 0)

If this is new to you, we recommend that you check out our zeros of polynomials article.

What you will learn in this lesson

While the

x

-intercepts are an important characteristic of the graph of a function, we need more in order to produce a good sketch.

Knowing the sign of a polynomial function between two zeros can help us fill in some of the gaps.

In this article, we'll learn how to determine the intervals over which a polynomial is positive or negative and connect this back to the graph.

Positive and negative intervals

The sign of a polynomial between any two consecutive zeros is either always positive or always negative.

For example, consider the graphed function

f (x) = (x + 1) (x - 1) (x - 3)

From the graph, we see that

f (x)

is always ...

...negative when $- \infty < x < - 1$ ‍.
...positive when $- 1 < x < 1$ ‍.
...negative when $1 < x < 3$ ‍.
...positive when $3 < x < \infty$ ‍.

It is not necessary, however, for a polynomial function to change signs between zeros.

For example, consider the graphed function

g (x) = x (x + 2)^{2}

From the graph, we see that

g (x)

is always...

...negative when $- \infty < x < - 2$ ‍.
...negative when $- 2 < x < 0$ ‍.
...positive when $0 < x < \infty$ ‍.

Notice that

g (x)

does not change sign around

x = - 2

Determining the positive and negative intervals of polynomials

Let's find the intervals for which the polynomial

f (x) = (x + 3) (x - 1)^{2}

is positive and the intervals for which it is negative.

The zeros of

f

are

- 3

and

1

. This creates three intervals over which the sign of

f

is constant:

Let’s find the sign of

f

for

- \infty < x < - 3

We know that

f

will either be always positive or always negative on this interval. We can determine which is the case by evaluating

f

for one value in this interval. Since

- 4

is in this interval, let's find

f (- 4)

Because we are only interested in the sign of the polynomial here, we don't have to completely evaluate it:

\begin{aligned} f (x) & = (x + 3) (x - 1)^{2} \\ f (- 4) & = (- 4 + 3) (- 4 - 1)^{2} \\ = (-) (-)^{2} & Evaluate only the sign of the answer. \\ = (-) (+) & A negative squared is a positive. \\ = - & A negative times a positive is a negative. \end{aligned}

Here we see that

f (- 4)

is negative, and so

f (x)

will always be negative for

- \infty < x < - 3

We can repeat the process for the remaining intervals.

Interval: $- 3 < x < 1$ ‍

Since

0

is in this interval, let's find the sign of

f (0)

$\begin{aligned} f (x) & = (x + 3) (x - 1)^{2} \\ f (0) & = (0 + 3) (0 - 1)^{2} \\ = (+) (-)^{2} & Evaluate only the sign of the answer. \\ = (+) (+) & A negative squared is a positive. \\ = + & A positive times a positive is a positive. \end{aligned}$ ‍

Because

f (0)

is positive,

f (x)

will be positive for

- 3 < x < 1

Interval: $1 < x < \infty$ ‍

Since

2

is in this interval, let's find the sign of

f (2)

$\begin{aligned} f (x) & = (x + 3) (x - 1)^{2} \\ f (2) & = (2 + 3) (2 - 1)^{2} \\ = (+) (+)^{2} & Evaluate only the sign of the answer. \\ = (+) (+) & A positive squared is a positive \\ = + & A positive times a positive is a positive. \end{aligned}$ ‍

Because

f (2)

is positive,

f (x)

will be positive for

1 < x < \infty

The results are summarized in the table below.

Interval	The value of a specific $f (x)$ ‍ within the interval	Sign of $f$ ‍ on interval	Connection to graph of $f$ ‍
$- \infty < x < - 3$ ‍	$f (- 4) < 0$ ‍	negative	Below the $x$ ‍-axis
$- 3 < x < 1$ ‍	$f (0) > 0$ ‍	positive	Above the $x$ ‍-axis
$1 < x < \infty$ ‍	$f (2) > 0$ ‍	positive	Above the $x$ ‍-axis

This is consistent with the graph of

y = f (x)

Check your understanding

g (x) = (x + 1)^{2} (x + 6)

has zeros at

x = - 6

and

x = - 1

What is the sign of $g$ ‍ on the interval $- 6 < x < - 1$ ‍?

h (x) = (3 - x) (x + 5) (x - 2)

has zeros at

x = - 5

x = 2

, and

x = 3

What is the sign of $h (x)$ ‍ on the interval $- 5 < x < 2$ ‍?

Challenge problem

3*) Which of the following could be the graph of $g (x) = (x - 2)^{2} (x + 1)^{3}$ ‍?

Determining positive & negative intervals from a sketch of the graph

Another way to determine the intervals over which a polynomial is positive or negative is to draw a sketch of its graph, based on the polynomial's end behavior and the multiplicities of its zeros.

Check out our graphs of polynomials article for further details.

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Staci
Posted 7 years ago. Direct link to Staci's post “Is there a section that c...”
Is there a section that covers finding the zeros of polynomials using p/q and synthetic division?
Button navigates to signup pageComment on Staci's post “Is there a section that c...”
(19 votes)
Answer
- Pieter
  Posted 5 years ago. Direct link to Pieter's post “They now have added a sec...”
  They now have added a section about synthetic division.
  Check it out here:
  https://www.khanacademy.org/math/algebra2/arithmetic-with-polynomials/synthetic-division-of-polynomials/v/synthetic-division
  
  I know my answer is a bit late but hopefully it may help others.
  
  Good luck!
  Button navigates to signup page
  (19 votes)
Felicia L.
Posted 7 years ago. Direct link to Felicia L.'s post “Once you know whether the...”
Once you know whether the first interval is positive or negative, isn't it easy to tell what the rest of the intervals are based on the multiplicity of the zeros? The graph will always either touch or cross the x axis at every zero.
Button navigates to signup pageComment on Felicia L.'s post “Once you know whether the...”
(18 votes)
Answer
- Jordan Cooper
  Posted 7 years ago. Direct link to Jordan Cooper's post “Correct. You could start...”
  Correct.
  
  You could start with any interval and work away from it. If an interval includes zero, I find that easiest to check in factored form. Or if you also have the unfactored polynomial, the intervals on the ends might be obvious based on the leading term.
  
  But the methods in this article work even if you haven't learned about the behavior of multiple roots yet.
  Button navigates to signup page
  (9 votes)
Sruja Konada
Posted 6 years ago. Direct link to Sruja Konada's post “im still confused as to h...”
im still confused as to how to find if the interval is positive or negative
Button navigates to signup pageButton navigates to signup page
(19 votes)
Answer
- WolfyNightshade
  Posted 3 months ago. Direct link to WolfyNightshade's post “Positive interval: The fu...”
  Positive interval: The function is above the x-axis
  Negative interval: The function is below the x-axis
  
  If you have a graph, look at the graph and see if the line for the function is above or below the x-axis.
  
  If you don't have a graph, then you need to test 1 or 2 values within the function.
  If Y is positive, then the interval is positive.
  If Y is negative, then the interval is negative.
  Button navigates to signup page
  (1 vote)
Dakota 9000
Posted 4 years ago. Direct link to Dakota 9000's post “all of this is total shen...”
all of this is total shenanigans in the first skill for positive and negative intervals for polynomials it says that an interval can be both positive and negative when it is clearly only one. In the hints, it even looks like they changed the interval to fit their answer. Please explain I've already watched the video for this and the article spending many hours trying to figure this out.
Button navigates to signup pageButton navigates to signup page
(10 votes)
Answer
- Kim Seidel
  Posted 4 years ago. Direct link to Kim Seidel's post “*Positive interval*: The...”
  Positive interval: The points for the function, or the graph sits above the x-axis
  Negative interval: The points for the function, or the graph sits below the x-axis
  
  If you have a graph, this is very easy - look at the graph and see if the line for the function sits above or below the x-axis.
  
  If you don't have a graph, then you need to test 1 or 2 values.
  -- If Y turns out to be positive, then the interval is positive because the point sits above the line.
  -- If Y turns out to be negative, then the interval is negative because the point sits below the line.
  
  For example, for the 1st problem to "Check Your Understanding", I used a value of x=-2 which is in the given interval. Plug in -2 for x in the function g(x)=(x+1)^2(x+6)
  g(-2) = (-2+1)^2(-2+6) = (-1)^2(4) = 1(4) = +4
  The interval is positive because the point (-2, 4) is above the x-axis.
  
  Hope this helps.
  Comment on Kim Seidel's post “*Positive interval*: The...”
  (6 votes)
Aunnesha
Posted 8 years ago. Direct link to Aunnesha's post “When doing problems with ...”
When doing problems with positive and negative intervals of polynomials and it asks what the sign of f(x) is, given only the zeros (not the graph itself), how do we know if the graph starts from below the point or above the point? Because the graph could be drawn two ways right?
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- Kim Seidel
  Posted 8 years ago. Direct link to Kim Seidel's post “Read thru this article ag...”
  Read thru this article again and try the practice problems. Sal gives an example that has no graphs. You basically have to test values that fall between the x-intercepts. And, as Sal shows above, you don't even need to do the complete math. Once you pick the number you want to test, you can just see what sign that creates.
  Button navigates to signup page
  (13 votes)
oliverhendleman
Posted 4 years ago. Direct link to oliverhendleman's post “Could you explain how to ...”
Could you explain how to solve this?

The polynomial p(x) = x^3 + 2x −11 has a real zero between which two consecutive integers?

(A) 0 and 1
(B) 1 and 2
(C) 2 and 3
(D) 3 and 4
(E) 4 and 5
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- loumast17
  Posted 4 years ago. Direct link to loumast17's post “You could graph it and lo...”
  You could graph it and look at where the zeroes look like they are. visually you could test each of the options.
  
  the alternative is to factor it and find the zeroes in factored form. This one really doesn't look like it factors nicely, so I think it wants you to just graph it.
  
  If you don't have a graphing calculator you can find something online. Desmos is really good.
  
  After you graph it check between each interval. look on the graph and see if the line crosses between 0 and 1, then 1 and 2 and so on.
  
  Let me know if you still can't quite get it.
  Button navigates to signup page
  (8 votes)
Tomatomath
Posted 3 years ago. Direct link to Tomatomath's post “i'm also confused by (no ...”
i'm also confused by (no breaks in the graph)as the meaning of
"continuous funtion"

can anybody explaine to me more detail by "no breaks in the graph"?
Button navigates to signup pageButton navigates to signup page
(5 votes)
Answer
- Kim Seidel
  Posted 3 years ago. Direct link to Kim Seidel's post “It is exactly what it say...”
  It is exactly what it says. You will have a line/curve that extends from the left side to the right side of the coordinate plane, and there will be no gaps. Or, you can think of the equation / function as being defined for inputs values (x) = all real numbers. No values are excluded. An exclusion would leave a gap.
  Button navigates to signup page
  (5 votes)
s_devin.beere
Posted 6 months ago. Direct link to s_devin.beere's post “I need help with everythi...”
I need help with everything right now.
Button navigates to signup pageComment on s_devin.beere's post “I need help with everythi...”
(7 votes)
Answer
keshav rawat
Posted 7 years ago. Direct link to keshav rawat's post “How do we figure out the ...”
How do we figure out the maximum or minimum point the graph of polynomial will take?
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- andrewp18
  Posted 7 years ago. Direct link to andrewp18's post “To do this for an arbitra...”
  To do this for an arbitrary polynomial, you need an understanding of calculus. I would suggest building up your trigonometry, geometry, and algebra background before diving into calculus!
  Button navigates to signup page
  (4 votes)
Sam D
Posted 7 years ago. Direct link to Sam D's post “The articles says in posi...”
The articles says in positive and negative intervals Why? which means that the only way to change signs is to cross the x-axis. But if this happened, the given zeros would not be consecutive! I get that to change the sign the function needs to cross the x axis, but what does the latter part mean? Not consecutive anymore?
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- Mikey Schwartz
  Posted 7 years ago. Direct link to Mikey Schwartz's post “What the article is sayin...”
  What the article is saying is this: A zero of a function is where the function touches the x axis. In order for the sign of a polynomial to change (positive to negative or negative to positive on the y axis), the polynomial needs to cross the x axis. Wherever the polynomial crosses the x axis, there will be a zero. Therefore, if two zeros are consecutive, the function cannot cross the x axis between them (because then there would be a zero in between those two zeros, which isn't possible because they are consecutive), and if the function cannot cross the x axis, the sign of the function cannot change.
  Comment on Mikey Schwartz's post “What the article is sayin...”
  (4 votes)